Assumption REF  was also used in [1]} to obtain the complimentary relationship with \(m\)  constant with source and drift terms. We will modify the proof in [1]} to obtain the optimal \(L^4(Q_T)\)  bounds on \(\nabla p_k\)  (see [3]} for an example of the optimality of the bounds), which will give us enough compactness to obtain the complimentary relationship.
